The problem of predicting the properties of glass materials has been a longstanding one in the field of glass and glass-ceramic chemistry. Because most glasses and glass-ceramics (hereinafter referred to collectively as “glasses”) contain a relatively large number of components, e.g., three to a half-a-dozen or more in many cases, the compositional space is multi-dimensional, making experimental study of the entire space economically impractical. Yet, from melting through to forming, the production of glass articles would clearly benefit from an ability to predict glass properties based on glass composition or, conversely, to select glass compositions based on desired properties.
Among all the technologically useful properties of a glass-forming system, the shear viscosity η of the melt is undoubtedly the most important. Every stage of industrial glass production—from the initial melting, mixing, and fining to the final forming operations—requires careful control of shear viscosity. For example, shear viscosity controls the rates of melting and of fining in a glass melting tank. Similarly, each glass forming operation, e.g., fiber forming or the final annealing of container glass, requires a certain well-defined viscosity range and consequently a specific temperature range for that operation. See, for example, Varshneya A K (2006) Fundamentals of Inorganic Glasses, 2nd ed. (Society of Glass Technology, Sheffield, UK). Viscosity also determines the relaxation rate of a final glass product. For example, viscosity controls the compaction behavior of display glasses (e.g., the glass sheets used as substrates in the production of liquid crystal displays) during customer heat treatment cycles. It should thus come as no surprise that the details of the viscosity-temperature relationship play a critical role in researching new glass compositions for display and other applications.
Among other reasons, the problem of relating viscosity to temperature and composition is challenging because from the initial glass melting to final forming, viscosity varies by over twelve orders of magnitude. See, for example, Varshneya (2006), supra. Viscosity is also sensitive to small changes in composition, especially in silicate melts where small levels of impurities can have a profound influence on the flow behavior. It is thus of great importance to have accurate knowledge of the scaling of viscosity with both composition (x) and temperature (T). Unfortunately, measurement of η(T,x) is challenging for high temperature melts, and low temperature measurements (i.e., in the high viscosity range, 1010 to 1015 Pa-s) are time consuming and often prohibitively expensive. See, for example, Varshneya (2006), supra. It is therefore of great interest to develop an accurate model of η(T,x).
Resistive furnaces require melts within a range of electrical resistivity values to ensure proper glass melting behavior and to avoid destruction of the tank refractory. The electrical resistivity of disordered media has drawn much interest from physicists due to the strong frequency dependence of the measured conductivity. See, for example, J. C. Dyre, P. Maass, B. Roling, and D. L. Sidebottom, “Fundamental Questions Relating to Ion Conduction in Disordered Solids,” Rep. Frog. Phys., 72, 046501 (2009). This frequency dependence is a direct result of inhomogeneities leading to a distribution of activation barriers for electrical conduction. While the universal frequency dependence of ac conductivity has received much attention, there has been little work addressing the temperature and composition dependences of conductivity at a fixed frequency. Most models assume an Arrhenius dependence of resistivity with temperature, despite the fact that as recognized as part of this disclosure, the same inhomogeneities that lead to a frequency-dependent conductivity must also lead to a non-Arrhenius dependence on temperature. As to the composition dependence of resistivity, the work that exists is based on strictly empirical fits, e.g., on Taylor series expansions of the coefficients of the Vogel-Fulcher-Tammann (VFT) relation. See, for example, O. V. Mazurin and O. A. Prokhorenko, “Electrical Conductivity of Glass Melts,” in Properties of Glass-Forming Melts, ed. by L. D. Pye, A. Montenero, and I. Joseph, pp. 295-338 (CRC Press, Taylor & Francis Group, Boca Raton, Fla., 2005); and A. Fluegel, D. A. Earl, and A. K. Varshneya, “Electrical Resistivity of Silicate Glass Melts Calculation Based on the SciGlass Database,” available online at http://glassproperties.com (2007).
Pavel Hrma of the Pacific Northwest National Laboratory (Richland, Wash.) reports an empirical model for the dependence of equilibrium viscosity as a function of temperature and composition. See P. Hrma, “Glass viscosity as a function of temperature and composition: A model based on Adam-Gibbs equation,” J. Non-Cryst. Solids, 354, 3389-3399 (2008). Hrma's model is based on the Adam-Gibbs equation, with the assumption in Hrma's Eq. (4) of a power law dependence for the configurational entropy. This assumption can lead to zero entropy (i.e., infinite viscosity) at a finite temperature, a physically dubious result. From a practical point of view, this means that viscosity predictions based on Hrma's model will suffer at low temperatures (i.e., high viscosities).
As to the composition dependence of viscosity, in Eqs. (8) and (9), Hrma includes composition dependence via linear expansions of the glass transition temperature and his s0 parameter in terms of the oxide components of the glass. However, as recognized as part of this disclosure, glass transition temperature cannot be expanded in such a manner over a wide range of compositions. For example, in borosilicate glasses the addition of sodium first causes a conversion of boron from three to four coordination, increasing the glass transition temperature. Then additional sodium creates non-bridging oxygens which subsequently decrease the glass transition temperature. Other examples include alkali or alkaline earth addition to aluminosilicate glasses and mixed alkali silicate glasses. Consequently, Hrma's linear expansion of the glass transition temperature is valid over only a narrow range of compositions. Hrma's second expansion is a linear expansion of his s0 parameter with respect to the oxide composition. As recognized as part of this disclosure, Hrma's expansion of s0 is analogous to an expansion of the T0 parameter in the VFT expansion. This is also unphysical. The result of this expansion is an overprediction of low temperature viscosities and an overprediction of fragility.
In view of this state of the art, a need exists for more effective methods and apparatus for predicting the properties of glass materials and, in particular, for predicting the dependence of viscosity and/or resistivity on temperature and/or composition. The present disclosure addresses these problems.